CONTINUUM MECHANICS ( TORSION as BOUNDARY VALUE PROBLEM - BVP )

1. CONTINUUM MECHANICS(TORSIONas BOUNDARY VALUE PROBLEM - BVP)

2. Problem formulation Body shape: straight prismatic bar with end surfaces perpendicular to the bar axis, cross-section of arbitrary shape. Loading: distributed loading over end surfaces yielding torque as only cross-sectional force, side surface free of loading, no volume forces. We will make also use of de Sain-Venant principle replacing distributed loading with a torque Kinematics boundary conditions: Bar fixed at one end (all displacements and their derivatives vanish there). In a further analysis we shall adopt the assumption of replacing kinematics conditions by statics ones (reaction torque); the bar is considered as being in the equilibrium but free to be twisted (free torsion). MS M MS=M

3. x2 A’ A x1 x3 x2 A’ A x1 Total twist angle r’  r    Assume: r’ = r 1  1 Assume: ? Twist angle per unit length (unit angle) Distortion function

4. - distortion function

5. 0 0 0 0 0 0 0 Laplacian or: The governing equation of torsion boundary value problem

6. x2 qν2 qν1 x1 Statics boundary conditions On a bar surface This is boundary value condition for distortion function differential equation On bar ends: + By de Saint Venant hypothesis MS JS Torsion inertia moment

7. x2 x1 2R Solid circular shaft Contour equation: Governing equation and boundary condition are homogeneous No distortion!

9. Solid circular shaft Polar inertia moment Unit twist angle r Twist angle Torsion section modulus Total twist angle for a shaft of length l

10. Rectangular bar h/2 max h/2 b For hb

11. Thin-walled bars Bars of closed cross-sections Bars of open cross-sections The behaviour of the above types of bars differs significantly when subjected to theaction of a torque. One can make a simple experiment cutting a tube:

12.  Closed thin-walled cross-sections Assume: prismatic tube of varying wall thickness Assume: constant distribution of shear stress across of tube thickness 1 1 2  2 From equilibrium condition: 

13. Closed thin-walled cross-sections S – area of the figure embedded within central curve s  ds dA  r(s) s

14. Open thin-walled cross-sections b2 A1 Assumptions: Cross-section partitioning: h1 A2 h2 b1 A3 b3 h3 A1 Solutions for torsion of rectangular bars obey: A2 A3

15. Open thin-walled cross-sections For hi/bi >6i =i

16. h/2 max max h/2 Open thin-walled cross-sections

17. Closed thin-walled cross-sections S – area of the figure embedded within central curve s  ds dA  r(s) s

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